MathDB

Problem 4

Part of 2001 Brazil Team Selection Test

Problems(2)

A, P, Q, O are concyclic if BP : PQ : QC = b : a : c

Source: brazilian tst 01

3/16/2005
Let ABCABC be a triangle with circumcenter OO. Let PP and QQ be points on the segments ABAB and ACAC, respectively, such that BP:PQ:QC=AC:CB:BABP : PQ : QC = AC : CB : BA. Prove that the points AA, PP, QQ and OO lie on one circle. Alternative formulation. Let OO be the center of the circumcircle of a triangle ABCABC. If PP and QQ are points on the sides ABAB and ACAC, respectively, satisfying BPPQ=CABC\frac{BP}{PQ}=\frac{CA}{BC} and CQPQ=ABBC\frac{CQ}{PQ}=\frac{AB}{BC}, then show that the points AA, PP, QQ and OO lie on one circle.
geometrycircumcirclegeometric transformationreflectiongeometry solved
modular congruence for sequence

Source: Brazil TST 2001 Test 2 P4

4/28/2021
Prove that for all integers n≥3n\ge3 there exists a set An={a1,a2,…,an}A_n=\{a_1,a_2,\ldots,a_n\} of nn distinct natural numbers such that, for each i=1,2,…,ni=1,2,\ldots,n, ∏1≤k≤nk≠iak≡1(modai).\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.
algebranumber theory